(1) Original BGK model:
The first existence result goes back to [1]} where Perthame established the existence of global weak solutions of the BGK model.
The uniqueness was guaranteed in a more stringent weighted \(L^{\infty }\)  space in [2]} (See [3]} for \(L^p\)  extension). The strong convergence to the Maxwellian is obtained by Desvillettes in [4]}. For the stationary case, Ukai constructs the existence theorem with a large boundary data in a 1-dimensional bounded interval.
The global-in-time classical solution of the BGK model near equilibrium can be found in [5]}, [6]}. For the particle system immersed in a fluid described by the coupled equation of the Navier-Stokes equation and
the BGK model, we refer to [7]}, [8]}, [9]}.
For the convergence analysis of numerical schemes of the BGK model, see [10]}, [11]}.
